Conway's topograph gives a geometric perspective of binary quadratic forms. Square form factorization is an integer factoring algorithm. We present a description of the algorithm using binary quadratic forms, and translate those steps in terms of the topograph. In particular, we illustrate the reduction operator and relate it to the quadratic progression around an infinity-gon in the topograph by using the connection between reduced forms and riverbends.

Excessive sun exposure is known to be the leading cause of skin cancer. The direct cellular damage inflicted by the ultraviolet (UV) radiation from the sun results in premature aging, DNA damage, and mutations that ultimately lead to skin cancer. Sunscreens are highly recommended to protect against UV radiation. However, little research has been conducted on the economic burden of sunscreen use. In this study, we aimed to evaluate the annual cost of sunscreen under both ideal and actual use conditions while stratifying for the sun protection factor (SPF) and by the name brand or equivalent store brand sunscreen. Pricing data was collected for sunscreens of SPF 30, 50, 70, and 100. For each type of sunscreen, the size and price of the container were recorded. Our results demonstrated that sunscreen prices increased with SPF but purchasing a generic sunscreen resulted in savings of 40%-50%. Our estimates reveal that sunscreens are affordable with annual expenditures ranging from $30.21 to $61.94, depending on brand, for SPF 50 sunscreens used with minimal application density for the average person.

Management of the Sacramento River and Sacramento–San Joaquin Delta (SRD) is one of California’s greatest challenges, requiring trade-offs between valued components that serve a multiplicity of conflicting purposes. Trade-offs do not signal a failure to create clever enough models, or scenarios that find a single optimal solution. Rather, an optimal solution that meets multiple objectives does not exist. We demonstrate an improved method for multiple-objective allocation of water: “turn-taking” optimization (TTO) within a multi-model cloud computing framework. We apply TTO to an array of physical hydrologic models that are linked with the Ecological Flows Tool (EFT): a multi-species decision support framework to evaluate how specific components of the flow regime promote and balance favorable habitat conditions for 15 representative species and 31 indicators within the SRD. Applying the TTO approach incorporates the existing modelled representation of socio-economic water management criteria, priorities, and constraints — and optimizes water-release patterns each water year using a dynamically shifting set of EFT indicators. Rather than attempting to optimize conditions for all ecological indicators every year, TTO creates flexibility and opportunities for different indicators to be successful in different years, informed by the frequency with which each species’ ecological needs should be met. As an individual EFT indicator is successful in a particular year, its priority in one or more subsequent years is reduced (and vice versa). Comparing TTO to a Reference Case scenario based on current management practices, 12 EFT indicators are improved, 14 show no change, and 5 show a reduction in suitability. When grouped into nine species and life-history groups, performance improved in four (late-fall-run Chinook, winter-run Chinook, spring-run Chinook, and Fremont cottonwood), did not change in four (fall-run Chinook Salmon, Delta Smelt, Splittail, and Longfin Smelt), and was worse in one group (Steelhead).

The use of botanical products and vitamins in skin care creams and sunscreens is prevalent. Herein we conduct an evaluation of sunscreens to quantitatively assess how often sunscreens incorporate botanically derived products and vitamins. The most commonly used botanicals products and vitamins are identified and stratified based on the sunscreen sun protection factor (SPF). The overall prevalence for the use of botanical agents and vitamins was 62% and 79%, respectively. Aloe vera and licorice root extracts were the most common botanical agents used in sunscreens. Retinyl palmitate was the most common vitamin derivative utilized in sunscreens. The prices of sunscreens significantly increased when more than one botanical product was added. Botanical products and vitamins are widely utilized in sunscreens and more research is needed to assess how their inclusion may enhance or alter the function of sunscreens.